City drew 1-1 with Real Madrid on Wednesday, confirming their exit from the group stage for the second year in a row.
Mancini's side have won none of their five games, and must beat group winners Borussia Dortmund in Germany to have any chance of qualifying for the Europa League.
However, Mancini said it was "crazy" to imagine his side could have won the competition.
Asked whether he could lose his job, he said. "No. I don't fear this.
"The season is long, you have Champions League, the FA Cup, Premier League,
"If we think we can win the Champions League after only two years we are crazy. We need to improve as a team, we did some mistakes. We are a good team. I don't have fear for this.
"Inside (the club) everyone knows football, and for us this is the second time (we've gone out of the group stage). To win it first you need to go into the second stage. After February everything can change."
Mancini pointed to Chelsea, who finally won the Champions League last season after nearly a decade of near misses.
He said: "Chelsea tried to win Champions League for 10 years. In the best moments, when Didier Drogba was young, they didn't win. They did it last year when no-one thought they could.
"We need to improve our team because there are a lot of teams better than us. I am not saying if you get to February you can't win. Anything can happen. But we didn't do that."
Real manager Jose Mourinho, who saw his side confirm their progress as runners-up, said he would be sacked for such a failure.
Mourinho, whose relationship with Mancini became strained after he replaced the Italian at Internazionale, said: "I admire the fact that clubs support their manager.
"City is out of Champions League. But Real Madrid and Borussia Dortmund are very good teams and we knew a big team would be out.
"It is good that it was City because Roberto can work without any kind of problem I believe.
"If it was Real Madrid, the press wouldn't let me return to Madrid."